Geometric Series — Practice Problems and Worked Solutions

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16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find the sum of the geometric series.

   $$\sum _{n=0}^{\infty }1{\left(\frac{1}{2}\right)}^n$$

   Answer: $2$

   Step-by-step solution

   1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
   2. $$\frac{1}{1-(\frac{1}{2})}=2$$
2. 2.

   Find the sum of the geometric series.

   $$\sum _{n=0}^{\infty }3{\left(\frac{1}{3}\right)}^n$$

   Answer: $\frac{9}{2}$

   Step-by-step solution

   1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
   2. $$\frac{3}{1-(\frac{1}{3})}=\frac{9}{2}$$
3. 3.

   Find the sum of the geometric series.

   $$\sum _{n=0}^{\infty }2{\left(-\frac{1}{2}\right)}^n$$

   Answer: $\frac{4}{3}$

   Step-by-step solution

   1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
   2. $$\frac{2}{1-(-\frac{1}{2})}=\frac{4}{3}$$
4. 4.

   Find the sum of the geometric series.

   $$\sum _{n=0}^{\infty }5{\left(\frac{2}{5}\right)}^n$$

   Answer: $\frac{25}{3}$

   Step-by-step solution

   1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
   2. $$\frac{5}{1-(\frac{2}{5})}=\frac{25}{3}$$
5. 5.

   Find the sum of the geometric series.

   $$\sum _{n=0}^{\infty }\frac{1}{2}{\left(\frac{3}{4}\right)}^n$$

   Answer: $2$

   Step-by-step solution

   1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
   2. $$\frac{\frac{1}{2}}{1-(\frac{3}{4})}=2$$
6. 6.

   Find the sum of the geometric series.

   $$\sum _{n=0}^{\infty }4{\left(-\frac{2}{3}\right)}^n$$

   Answer: $\frac{12}{5}$

   Step-by-step solution

   1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
   2. $$\frac{4}{1-(-\frac{2}{3})}=\frac{12}{5}$$
7. 7.

   Find the sum of the geometric series.

   $$\sum _{n=0}^{\infty }2{\left(\frac{1}{4}\right)}^n$$

   Answer: $\frac{8}{3}$

   Step-by-step solution

   1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
   2. $$\frac{2}{1-(\frac{1}{4})}=\frac{8}{3}$$
8. 8.

   Find the sum of the geometric series.

   $$\sum _{n=0}^{\infty }6{\left(\frac{1}{3}\right)}^n$$

   Answer: $9$

   Step-by-step solution

   1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
   2. $$\frac{6}{1-(\frac{1}{3})}=9$$
9. 9.

   Find the sum of the geometric series.

   $$\sum _{n=0}^{\infty }1{\left(-\frac{1}{3}\right)}^n$$

   Answer: $\frac{3}{4}$

   Step-by-step solution

   1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
   2. $$\frac{1}{1-(-\frac{1}{3})}=\frac{3}{4}$$
10. 10.

    Find the sum of the geometric series.

    $$\sum _{n=0}^{\infty }3{\left(\frac{2}{3}\right)}^n$$

    Answer: $9$

    Step-by-step solution

    1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
    2. $$\frac{3}{1-(\frac{2}{3})}=9$$
11. 11.

    Find the sum of the geometric series.

    $$\sum _{n=0}^{\infty }5{\left(\frac{1}{5}\right)}^n$$

    Answer: $\frac{25}{4}$

    Step-by-step solution

    1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
    2. $$\frac{5}{1-(\frac{1}{5})}=\frac{25}{4}$$
12. 12.

    Find the sum of the geometric series.

    $$\sum _{n=0}^{\infty }2{\left(\frac{3}{5}\right)}^n$$

    Answer: $5$

    Step-by-step solution

    1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
    2. $$\frac{2}{1-(\frac{3}{5})}=5$$
13. 13.

    Find the sum of the geometric series.

    $$\sum _{n=0}^{\infty }8{\left(-\frac{1}{2}\right)}^n$$

    Answer: $\frac{16}{3}$

    Step-by-step solution

    1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
    2. $$\frac{8}{1-(-\frac{1}{2})}=\frac{16}{3}$$
14. 14.

    Find the sum of the geometric series.

    $$\sum _{n=0}^{\infty }1{\left(\frac{4}{5}\right)}^n$$

    Answer: $5$

    Step-by-step solution

    1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
    2. $$\frac{1}{1-(\frac{4}{5})}=5$$
15. 15.

    Find the sum of the geometric series.

    $$\sum _{n=0}^{\infty }7{\left(-\frac{1}{4}\right)}^n$$

    Answer: $\frac{28}{5}$

    Step-by-step solution

    1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
    2. $$\frac{7}{1-(-\frac{1}{4})}=\frac{28}{5}$$
16. 16.

    Find the sum of the geometric series.

    $$\sum _{n=0}^{\infty }\frac{1}{3}{\left(\frac{1}{2}\right)}^n$$

    Answer: $\frac{2}{3}$

    Step-by-step solution

    1. $$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}(|r|<1)$$
    2. $$\frac{\frac{1}{3}}{1-(\frac{1}{2})}=\frac{2}{3}$$

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• Improper Integrals
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• Convergence Tests
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• Taylor Series
• Parametric Derivatives
• Parametric Arc Length
• Polar Area